# The Method of Direct Integration Examples 2

Recall from The Method of Direct Integration page that if we have a differential equation in the form $\frac{dy}{dt} = ay + b$ where $a$ and $b$ are constants, then the general solution can be obtained by rewriting this differential equation as to get the lefthand side to be the derivative of a natural logarithm while in general, the solution is given as:

(1)Let's look at some examples of solving differential equations by direct integration.

## Example 1

**Find all solutions to the differential equation $\frac{dy}{dt} = 11y + 132$.**

We will first rewrite out differential equation and solve as follows:

(2)Note that if $D = 0$ then we get $y = 12$ which is a solution to our differential equation, and so all solutions are given by $y = De^{11t} - 12$ for $D$ as a constant.

## Example 2

**Find all solutions to the differental equation $y \frac{dy}{dt} - 4y^2 + 36y = 0$.**

If we rewrite the differential equation above as $\frac{dy}{dt} = 4y - 36$ by dividing all terms by $y$ (provided $y \neq 0$) and then rearranging terms. Thus we solve this differential equation as:

(3)Note that if $D = 0$ we get that $y = 9$ which is a solution to our differential equation. Also, $y = 0$ (which we initially omitted) is also a solution to our differential equation. Thus all solutions are given by $y = De^{4t} + 9$ for $D$ as a constant, and $y = 0$.